Theorem clwwlknun 16311

Description: The set of closed walks of fixed length N in a simple graph G is the union of the closed walks of the fixed length N on each of the vertices of graph G. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)

Hypothesis

Ref	Expression
clwwlknun.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
clwwlknun	|- ( G e. USGraph -> ( N ClWWalksN G ) = U_ x e. V ( x (ClWWalksNOn ` G ) N ) )

Distinct variable groups:   x, G   x, N   x, V

Proof of Theorem clwwlknun

Dummy variables y i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 3974	. . 3 |- ( y e. U_ x e. V ( x (ClWWalksNOn ` G ) N ) <-> E. x e. V y e. ( x (ClWWalksNOn ` G ) N ) )
2		isclwwlknon 16300	. . . . 5 |- ( y e. ( x (ClWWalksNOn ` G ) N ) <-> ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) )
3	2	rexbii 2539	. . . 4 |- ( E. x e. V y e. ( x (ClWWalksNOn ` G ) N ) <-> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) )
4		simpl 109	. . . . . 6 |- ( ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) -> y e. ( N ClWWalksN G ) )
5	4	rexlimivw 2646	. . . . 5 |- ( E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) -> y e. ( N ClWWalksN G ) )
6		clwwlknun.v	. . . . . . . . 9 |- V = (Vtx ` G )
7		eqid 2231	. . . . . . . . 9 |- (Edg ` G ) = (Edg ` G )
8	6, 7	clwwlknp 16287	. . . . . . . 8 |- ( y e. ( N ClWWalksN G ) -> ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) )
9	8	anim2i 342	. . . . . . 7 |- ( ( G e. USGraph /\ y e. ( N ClWWalksN G ) ) -> ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) )
10	7, 6	usgrpredgv 16068	. . . . . . . . . . . . 13 |- ( ( G e. USGraph /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) -> ( (lastS ` y ) e. V /\ ( y ` 0 ) e. V ) )
11	10	ex 115	. . . . . . . . . . . 12 |- ( G e. USGraph -> ( { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) -> ( (lastS ` y ) e. V /\ ( y ` 0 ) e. V ) ) )
12		simpr 110	. . . . . . . . . . . 12 |- ( ( (lastS ` y ) e. V /\ ( y ` 0 ) e. V ) -> ( y ` 0 ) e. V )
13	11, 12	syl6com 35	. . . . . . . . . . 11 |- ( { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) -> ( G e. USGraph -> ( y ` 0 ) e. V ) )
14	13	3ad2ant3 1046	. . . . . . . . . 10 |- ( ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) -> ( G e. USGraph -> ( y ` 0 ) e. V ) )
15	14	impcom 125	. . . . . . . . 9 |- ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) -> ( y ` 0 ) e. V )
16		simpr 110	. . . . . . . . . . . 12 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> x = ( y ` 0 ) )
17	16	eqcomd 2237	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> ( y ` 0 ) = x )
18	17	biantrud 304	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> ( y e. ( N ClWWalksN G ) <-> ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
19	18	bicomd 141	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> ( ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) <-> y e. ( N ClWWalksN G ) ) )
20	15, 19	rspcedv 2914	. . . . . . . 8 |- ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) -> ( y e. ( N ClWWalksN G ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
21	20	adantld 278	. . . . . . 7 |- ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) -> ( ( G e. USGraph /\ y e. ( N ClWWalksN G ) ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
22	9, 21	mpcom 36	. . . . . 6 |- ( ( G e. USGraph /\ y e. ( N ClWWalksN G ) ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) )
23	22	ex 115	. . . . 5 |- ( G e. USGraph -> ( y e. ( N ClWWalksN G ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
24	5, 23	impbid2 143	. . . 4 |- ( G e. USGraph -> ( E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) <-> y e. ( N ClWWalksN G ) ) )
25	3, 24	bitrid 192	. . 3 |- ( G e. USGraph -> ( E. x e. V y e. ( x (ClWWalksNOn ` G ) N ) <-> y e. ( N ClWWalksN G ) ) )
26	1, 25	bitr2id 193	. 2 |- ( G e. USGraph -> ( y e. ( N ClWWalksN G ) <-> y e. U_ x e. V ( x (ClWWalksNOn ` G ) N ) ) )
27	26	eqrdv 2229	1 |- ( G e. USGraph -> ( N ClWWalksN G ) = U_ x e. V ( x (ClWWalksNOn ` G ) N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   {cpr 3670   U_ciun 3970   ` cfv 5326  (class class class)co 6018   0cc0 8032   1c1 8033   + caddc 8035   - cmin 8350  ..^cfzo 10377  ♯chash 11038  Word cword 11117  lastSclsw 11162  Vtxcvtx 15882  Edgcedg 15927  USGraphcusgr 16024   ClWWalksN cclwwlkn 16273  ClWWalksNOncclwwlknon 16296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-2o 6583  df-er 6702  df-map 6819  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11118  df-ndx 13103  df-slot 13104  df-base 13106  df-edgf 15875  df-vtx 15884  df-iedg 15885  df-edg 15928  df-umgren 15964  df-usgren 16026  df-clwwlk 16262  df-clwwlkn 16274  df-clwwlknon 16297

This theorem is referenced by: (None)